8.1.8.1.

Competing risk model

Use the competing risk model when the failure mechanisms are independent
and the first mechanism failure causes the component to fail

Assume a (replaceable) component or unit has \(k\)
different ways it can fail. These are called failure modes
and underlying each failure mode is a failure mechanism.

The Competing Risk Model evaluates component reliability by "building
up" from the reliability models for each failure mode.

The following three assumptions are needed.

Each failure mechanism leading to a particular type of failure (i.e., failure
mode) proceeds independently of every other one, at least until a failure
occurs.

The component fails when the first of all the competing failure
mechanisms reaches a failure state.

Each of the \(k\)
failure modes has a known life distribution model \(F_i(t)\).

The competing risk model can be used when all three assumptions hold. If
\(R_c(t)\), \(F_c(t)\), and \(h_c(t)\)
denote the reliability, CDF and failure rate for the component, respectively, and
\(R_i(t)\), \(F_i(t)\), and \(h_i(t)\)
are the reliability, CDF and failure rate for the \(i\)-th
failure mode, respectively, then the competing risk model formulas are:

All the failure mechanisms are having a race to see which can
reach failure first. They are not allowed to "look over their shoulder
or sideways" at the progress the other ones are making. They just go their
own way as fast as they can and the first to reach "failure" causes the
component to fail.

Under these conditions the component reliability is the product of the
failure mode reliabilities and the component failure rate is just the sum
of the failure mode failure rates.

Note that the above holds for any arbitrary life distribution model, as
long as "independence" and "first mechanism failure causes the component
to fail" holds.

When we learn how to plot and analyze reliability data in later sections,
the methods will be applied separately to each failure mode within the
data set (considering failures due to all other modes as "censored
run times"). With this approach, the competing risk model provides
the glue to put the pieces back together again.